Critical phenomena in gravitational collapse

arXiv:gr-qc/9712084v3 27 Jan 1998

Critical phenomena in gravitational collapse Carsten Gundlach Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut) Schlaatzweg 1, 14473 Potsdam, Germany 9 Jan 1998, revised 26 Jan 1998 Abstract As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, power-law scaling of the black hole mass, and scale echoing have given rise to the term “critical phenomena”. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically self-similar. This review gives an introduction to the phenomena, tries to summarize the essential features of what is happening, and then presents extensions and applications of this basic scenario. Critical phenomena are of interest particularly for creating surprising structure from simple equations, and for the light they throw on cosmic censorship. They may have applications in quantum black holes and astrophysics.

Contents 1 Introduction

2

2 A look at the phenomena 2.1 The spherically symmetric scalar field . . . . . . . . . . . . . . . 2.2 The spherically symmetric perfect fluid . . . . . . . . . . . . . . .

4 4 8

3 The 3.1 3.2 3.3

basic scenario Scale-invariance, self-similarity, and homothety . . . . . . . . . . Gravity regularizes self-similar matter . . . . . . . . . . . . . . . Universality and scaling . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 15

4 Extensions of the basic scenario 17 4.1 More matter models . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 CSS and DSS critical solutions . . . . . . . . . . . . . . . . . . . 18 1

4.3 4.4 4.5

4.6 4.7

Black hole thresholds with a mass gap . . . . . . . Approximate self-similarity and universality classes Beyond spherical symmetry . . . . . . . . . . . . . 4.5.1 Axisymmetric gravitational waves . . . . . 4.5.2 Perturbing around sphericity . . . . . . . . Critical phenomena and naked singularities . . . . Black hole charge and angular momentum . . . . . 4.7.1 Charge . . . . . . . . . . . . . . . . . . . . 4.7.2 Angular momentum . . . . . . . . . . . . .

5 More speculative aspects 5.1 The renormalisation group as a time evolution 5.2 Analytic approaches . . . . . . . . . . . . . . . 5.3 Astrophysical applications? . . . . . . . . . . . 5.4 Critical collapse in semiclassical gravity . . . . 6 Summary and conclusions

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Introduction

In 1987 Christodoulou, who was then (and still is) studying the spherically symmetric Einstein-scalar model analytically [32, 33, 34, 35, 36] suggested to Matt Choptuik, who was investigating the same system numerically, the following question [26]: Consider a generic smooth one-parameter family of initial data, such that for large values of the parameter p a black hole is formed, and no black hole is formed for small p. If one makes a bisection search for the critical value p∗ where a black hole is just formed, does it have finite or infinitesimal mass? After developing advanced numerical methods for this purpose, Choptuik managed to give highly convincing numerical evidence that the mass is infinitesimal. Moreover he found two totally unexpected phenomena: The first is the now famous scaling relation M ≃ C (p − p∗ )γ

(1)

for the black hole mass M in the limit p ≃ p∗ (but p > p∗ ), where the constant γ is the same for all such one-parameter families. (Choptuik found γ ≃ 0.37.) The second is the appearance of a highly complicated, scale-periodic solution for p ≃ p∗ , which is again the same for all initial data as long as they are near the limit of black hole formation. The logarithmic scale period of this solution, ∆ ≃ 3.44, is a second dimensionless number coming out of the blue. Until then most relativists would have assigned numerical work the role of providing quantitative details of phenomena that were already understood qualitatively, noticeably in astrophysical applications. Here, numerical relativity provided an important qualitative input into mathematical relativity and 2

gave rise to a new research field. Similar phenomena to Choptuik’s results were quickly found in other systems too, suggesting that they were limited neither to scalar field matter nor to spherical symmetry. Many researchers were intrigued by the appearance of a complicated “echoing” structure, and the two mysterious dimensionless numbers, in the critical solution. Later it was realized that critical phenomena also provide a natural route to naked singularities, and this has linked critical phenomena to the mainstream of research in mathematical relativity. Purely analytical approaches, however, have not been successful so far, and most of what is understood in critical phenomena is based on a mixture of analytical and numerical work. Scale-invariance, universality and power-law behavior suggest the name critical phenomena. A connection with the renormalisation group in partial differential equations has been established in hindsight, but has not yet provided fresh input. The connection with the renormalisation group in statistical mechanics is even more tenuous, limited to approximate scale invariance, but not extending to the presence of a statistical ensemble. In our presentation we combine a phenomenological with a systematic approach. In order to give the reader not familiar with Choptuik’s work a flavor of how complicated phenomena arise from innocent-looking PDEs, we describe his results in some detail, followed by a review of the work of Coleman and Evans on critical phenomena in perfect fluid collapse, which appeared a year later. (The important paper of Abrahams and Evans, historically the first paper after Choptuik’s, is reviewed in the context of non-spherically symmetric systems.) After this phenomenological opening, we systematically explain the key features echoing, universality and scaling in a coherent scenario which has emerged over time, with key terminology borrowed from dynamical systems and renormalisation group theory. This picture is partly qualitative, but has been underpinned by successful semi-analytic calculations of Choptuik’s (and other) critical solutions and the critical exponent γ to high precision. Semi-analytic here means that although an analytic solution is impossible, the numerical work solves a simplified problem, for example reducing a PDE to an ODE. In this context we introduce the relativistic notions of scale-invariance and scale-periodicity, define the concept of a critical solution, and sketch the calculation of the critical exponent. In the following section we present extensions of this basic scenario. This presentation is again systematic, but to also give the phenomenological point of view, the section starts with a tabular overview of the matter models in which critical phenomena have been studied so far. Extensions of the basic scenario include more realistic matter models, critical phenomena with a mass gap, the study of the global structure of the critical spacetime itself, and black holes with charge and mass. In a final section that could be titled “loose ends”, we group together approaches to the problem that have failed or are as yet at a more speculative stage. This section also reviews some detailed work on the quantum aspects of 3

critical collapse, based on various toy models of semiclassical gravity. Previous short review papers include Horne [65], Bizo´ n [9] and Gundlach [49]. Choptuik is preparing a longer review paper [28]. For an interesting general review of the physics of scale-invariance, see [95].

2 2.1

A look at the phenomena The spherically symmetric scalar field

The system in which Christodoulou and Choptuik studied gravitational collapse in detail was the spherically symmetric massless, minimally coupled scalar field. It has the advantage of simplicity, and the scalar radiation propagating at the speed of light mimics gravitational waves within spherical symmetry. The Einstein equations are   1 Gab = 8π ∇a φ∇b φ − gab ∇c φ∇c φ (2) 2 and the matter equation is ∇a ∇a φ = 0.

(3)

Note that the matter equation of motion is contained within the contracted Bianchi identities. Choptuik chose Schwarzschild-like coordinates ds2 = −α2 (r, t) dt2 + a2 (r, t) dr2 + r2 dΩ2 ,

(4)

where dΩ2 = dθ2 + sin2 θ dϕ2 is the metric on the unit 2-sphere. This choice of coordinates is defined by the radius r giving the surface of 2-spheres as 4πr2 , and by t being orthogonal to r. One more condition is required to fix the coordinate completely. Choptuik chose α = 1 at r = 0, so that t is the proper time of the central observer. In the auxiliary variables Φ = φ,r ,

Π=

a φ,t , α

the wave equation becomes a first-order system, α  Π , Φ,t = a ,r  1 α  Π,t = Φ . r2 a ,r

(5)

(6) (7)

In spherical symmetry there are four algebraically independent components of the Einstein equations. Of these, one is proportional to derivatives of the other and can be disregarded. The other three contain only first derivatives of the metric, namely a,t , a,r and α,r . Choptuik chose to use the equations giving a,r 4

and α,r for his numerical scheme, so that only the scalar field is evolved, but the two metric coefficients are calculated from the matter at each new time step. (The main advantage of such a numerical scheme is its stability.) These two equations are 1 a2 − 1 a,r + − 2πr(Π2 + Φ2 ) = a 2r 1 a2 − 1 1 α,r − a,r − = α a 2r

0,

(8)

0,

(9)

and they are, respectively, the Hamiltonian constraint and the slicing condition. These four first-order equations totally describe the system. For completeness, we also give the remaining Einstein equation, 1 a,t = 2πr(Π2 − Φ2 ). α

(10)

The free data for the system are the two functions Π(r) and Φ(r). (In spherical symmetry, there are no physical degrees of freedom in the gravitational field.) Choptuik investigated many one-parameter families of such data by evolving the data for many values each of the parameter, say p. Simple examples of such families are Φ(r) = 0 and a Gaussian for Π(r), with the parameter p taken to be either the amplitude of the Gaussian, with the width and center fixed, or the width, with position and amplitude fixed, or the position, with width and amplitude fixed. It is plausible that for the amplitude sufficiently small, with width and center fixed, the scalar field will disperse, and for sufficiently large amplitude will form a black hole, with similar behavior for many generic parameters. This is difficult to prove in generality. Christodoulou showed for the spherically symmetric scalar field system that data sufficiently weak in a well-defined way evolve to a Minkowski-like spacetime [34], and that a class of sufficiently strong data forms a black hole [33]. But what happens in between? Choptuik found that in all families of initial data he could make arbitrarily small black holes by fine-tuning the parameter p close to the black hole threshold. An important fact is that there is nothing visibly special to the black hole threshold. One cannot tell that one given data set will form a black hole and another one infinitesimally close will not, short of evolving both for a sufficiently long time. Fine-tuning is then a heuristic procedure, and effectively proceeds by bisection: Starting with two data sets one of which forms a black hole, try a third one in between along some oneparameter family linking the two, drop one of the old sets and repeat. With p closer to p∗ , the spacetime varies on ever smaller scales. The only limit was numerical resolution, and in order to push that limitation further away, Choptuik developed special numerical techniques that recursively refine the numerical grid in spacetime regions where details arise on scales too small to be resolved properly. In the end, Choptuik could determine p∗ up to a relative 5

precision of 10−15 , and make black holes as small as 10−6 times the ADM mass of the spacetime. The power-law scaling (1) was obeyed from those smallest masses up to black hole masses of, for some families, 0.9 of the ADM mass, that is, over six orders of magnitude [26]. There were no families of initial data which did not show the universal critical solution and critical exponent. Choptuik therefore conjectured that γ is the same for all one-parameter families, and that the approximate scaling law holds ever better for arbitrarily small p − p∗ . I would suggest reformulating this conjecture in a different manner. Let us first consider a finite-dimensional subspace of the space of initial data, with coordinates pi on it. The subspace of Gaussian data for both Φ and Π for example is 6-dimensional. We could choose the amplitudes, centers and widths of the two Gaussians as coordinates, but any six smooth functions of these could also serve as coordinates. Various one-parameter families only serve as probes of this one 6-dimensional space. They indicate that there is a smooth hypersurface in this space which divides black hole from non-black hole data. Let P (pi ) be any smooth coordinate function on the space so that P (pi ) = 0 is the black hole threshold. Then, for any choice of P (pi ), there is a second smooth function C(pi ) on the space so that the black hole mass as a function of the space is given as M = θ(P ) CP γ . (11) In words, the entire unsmoothness at the black hole threshold is captured by the one critical exponent. One can now formally go over from a finite-dimensional subspace to the infinite-dimensional space of initial data. Note that Φ and Π must be square-integrable for the spacetime to be asymptotically flat, and therefore the initial data space has a countable basis. In this view, it is difficult to see how different one-parameter families could have different values of γ. It also shows that the critical exponent is not an effect of a bad parameterization. Clearly a collapse spacetime which has ADM mass 1, but settles down to a black hole of mass (for example) 10−6 has to show structure on very different scales. The same is true for a spacetime which is as close to the black hole threshold, but on the other side: the scalar wave contracts until curvature values of order 1012 are reached in a spacetime region of size 10−6 before it starts to disperse. Choptuik found that all near-critical spacetimes, for all families of initial data, look the same in an intermediate region, that is they approximate one universal spacetime, which is also called the critical solution. This spacetime is scale-periodic in the sense that there is a value t∗ of t such that when we shift the origin of t to t∗ , we have
Z(r, t) = Z en∆ r, en∆ t (12)

for all integer n and for ∆ ≃ 3.44, and where Z stands for any one of a, α or φ (and therefore also for rΠ or rΦ). The accumulation point t∗ depends on the family, but the scale-periodic part of the near-critical solutions does not.

6

This result is sufficiently surprising to formulate it once more in a slightly different manner. Let us replace r and t by a pair of auxiliary variables such that one of them retains a dimension, while the other is dimensionless. A simple example is (after shifting the origin of t to t∗ )   r t x = − , τ = − ln − , t < 0. (13) t L (As a matter of convention, t has been assumed negative so that it increases towards the accumulation point at t = r = 0. Similarly, τ has been defined so that it increases with increasing t.) Choptuik’s observation, expressed in these coordinates, is that in any near-critical solution there is a space-time region where the fields a, α and φ are well approximated by their values in a universal solution, as Z(x, τ ) ≃ Z∗ (x, τ ), (14) where the fields a∗ , α∗ and φ∗ of the critical solution have the property Z∗ (x, τ + ∆) = Z∗ (x, τ ).

(15)

The dimensionful constants t∗ and L depend on the one-parameter family of solutions, but the dimensionless critical fields a∗ , α∗ and φ∗ , and in particular their dimensionless period ∆, are universal. A slightly supercritical and a slightly subcritical solution from the same family (so that L and t∗ are the same) are practically indistinguishable until they have reached a very small scale where the one forms an apparent horizon, while the other starts dispersing. Not surprisingly, this scale is the same as that of the black hole (if one is formed), and so we have for the range ∆τ of τ on which a near-critical solution approximates the universal one ∆τ ≃ γ ln |p − p∗ | + const (16) and for the number N of scaling “echos” that are seen, N ≃ ∆−1 γ ln |p − p∗ | + const.

(17)

Note that this holds for both supercritical and subcritical solutions. Choptuik’s results have been repeated by a number of other authors. Gundlach, Price and Pullin [56] could verify the mass scaling law with a relatively simple code, due to the fact that it holds even quite far from criticality. Garfinkle [44] used the fact that recursive grid refinement in near-critical solutions is not required in arbitrary places, but that all refined grids are centered on (r = 0, t = t∗ ), in order to simulate a simple kind of mesh refinement on a single grid in double null coordinates: u grid lines accumulate at u = 0, and v lines at v = 0, with (v = 0, u = 0) chosen to coincide with (r = 0, t = t∗ ). Hamad´e and Stewart [58] have written a complete mesh refinement algorithm based on a double null grid (but coordinates u and r), and report even higher resolution than Choptuik. Their coordinate choice also allowed them to follow the evolution beyond the formation of an apparent horizon. 7

2.2

The spherically symmetric perfect fluid

The scale-periodicity, or echoing, of the scalar field critical solution was a new phenomenon in general relativity, and initial efforts at understanding concentrated there. Evans however realized that scale-echoing was only a more complicated form of scale-invariance, that the latter was the key to the problem, and moreover that it could be expected to arise in a different matter model, namely a perfect fluid. Evans knew that scale-invariant, or self-similar, solutions arise in fluid dynamics problems (without gravity) when there are two very different scales in the initial problem (for example an explosion with high initial density into a thin surrounding fluid [5]), and that such solutions play the role of an intermediate asymptotic in the intermediate density regime [6]. Evans and Coleman [41] therefore made a perfect fluid ansatz for the matter, Gab = 8π [(p + ρ)ua ub + pga b]

(18)

where ua is the 4-velocity, ρ the density and p the pressure. As for the scalar field, the matter equations of motion are equivalent to the conservation of matter energy-momentum. The only equation of state without an intrinsic scale is p = kρ, with k a constant. This was desirable in order to allow for a scaleinvariant solution like that of Choptuik. Evans and Coleman chose k = 1/3 because it is the equation of state of radiation (or ultra-relativistic hot matter). They made the same coordinate choice in spherical symmetry as Choptuik, and evolved one-parameter families of initial data. They found a universal intermediate attractor, and power-law scaling of the black hole mass, with a universal critical exponent of γ ≃ 0.36. (To anticipate, the coincidence of the value with that for the scalar field is now believed to be accidental.) The main difference was that the universal solution is not scale-periodic but scaleinvariant: it has the continuous symmetry (after shifting the origin of t to t∗ )  r = Z(x) (19) Z∗ (r, t) = Z∗ − t

where Z now stands for the metric coefficients a and α (as in the scalar field case) and the dimensionless matter variables t2 ρ and ur . We shall discuss this symmetry in more detail below. Independently, Evans and Coleman made a scale-invariant ansatz for the critical solution, which transforms the PDE problem in t and r into an ODE problem in the one independent variable x. They then posed a nonlinear boundary value problem by demanding regularity at the center x = 0 and at the past sound cone x = x0 of the point (t = r = 0), where a generic self-similar solution would be singular. The sound cone referred to here and below is a characteristic of the matter equations of motion. It is made up of the characteristic curves which are also homothetic curves (x = const.). The past light cone of (t = r = 0) plays no role in the spherically symmetric perfect fluid critical solution because in spherical symmetry there are no propagating gravitational 8

degrees of freedom. It does play a role in the spherically symmetric scalar field critical solutions however, because there it is also characteristic of the scalar field matter. The regularity condition at the center x = 0 is local flatness, or a = 1 in the coordinates (4). The regularity condition at x = x0 is the absence of a shock wave. I believe that both conditions are equivalent to demanding analyticity, and that x = 0 and x = x0 are “regular singular points” of the ODE system, although this remains to be shown by a suitable change of variables. The solutions of this boundary value problem form a discrete family. The simplest solution of the boundary value problem coincides perfectly with the intermediate asymptotic that is found in the collapse simulations, arising from generic data. It is really this coincidence that justifies the boundary value problem posed by Evans and Coleman. At an intuitive level, however, one could argue that the critical solution should be smooth because it arises as an intermediate asymptotic from smooth initial data. In a contrasting opinion, Carr and Henriksen [20] claim that the perfect fluid critical solution should obey a certain global condition (the “particle horizon” and the “event horizon” of the spacetime coincide) that can be interpreted as the solution being a marginal black hole. In order to impose this condition, they need one more free parameter in the space of CSS solutions, and obtain it by not imposing analyticity at the past sound cone, where their candidate critical solution has a shock. Evans and Coleman also suggested that an analysis of the linear perturbations of the critical solution would give an “estimate” of the critical exponents. This program was carried out for the k = 1/3 perfect fluid by Koike, Hara and Adachi [71] and for other values of k by Maison [75], to high precision.

3 3.1

The basic scenario Scale-invariance, self-similarity, and homothety

The critical solution found by Choptuik [24, 25, 26] for the spherically symmetric scalar field is scale-periodic, or discretely self-similar (DSS), and the critical solution found by Evans and Coleman [41] is scale-invariant, or continuously self-similar (CSS). We begin with the continuous symmetry because it is simpler. In Newtonian physics, a solution Z is self-similar if it is of the form   ~x (20) Z(~x, t) = Z f (t) If the function f (t) is derived from dimensional considerations √ alone, one speaks of self-similarity of the first kind. An example is f (t) = λt for the diffusion equation Z,t = λZ,xx . In more complicated equations, the limit of self-similar solutions can be singular, and f (t) may contain additional dimensionful constants (which do not appear in the field equation) in terms such as (t/L)α , 9

where α, called an anomalous dimension, is not determined by dimensional considerations but through the solution of an eigenvalue problem [6]. For now, we concentrate on self-similarity of the first kind. A continuous self-similarity of the spacetime in GR corresponds to the existence of a homothetic vector field ξ, defined by the property [18] Lξ gab = 2gab .

(21)

(This is a special type of conformal Killing vector, namely one with constant coefficient on the right-hand side. The value of this constant coefficient is conventional, and can be set equal to 2 by a constant rescaling of ξ.) From (21) it follows that Lξ Ra bcd = 0, (22) and therefore Lξ Gab = 0,

(23)

but the inverse does not hold: the Riemann tensor and the metric need not satisfy (22) and (21) if the Einstein tensor obeys (23). If the matter is a perfect fluid (18) it follows from (21), (23) and the Einstein equations that Lξ ua = −ua ,

Lξ ρ = −2ρ,

Lξ p = −2p.

(24)

Similarly, if the matter is a free scalar field φ (2), it follows that Lξ φ = κ,

(25)

where κ is a constant. In coordinates xµ = (τ, xi ) adapted to the homothety, the metric coefficients are of the form gµν (τ, xi ) = e−2τ g˜µν (xi ) (26) where the coordinate τ is the negative logarithm of a spacetime scale, and the remaining three coordinates xi are dimensionless. In these coordinates, the homothetic vector field is ∂ ξ=− . (27) ∂τ The minus sign in both equations (26) and (27) is a convention we have chosen so that τ increases towards smaller spacetime scales. For the critical solutions of gravitational collapse, we shall later choose surfaces of constant τ to be spacelike (although this is not possible globally), so that τ is the time coordinate as well as the scale coordinate. Then it is natural that τ increases towards the future, that is towards smaller scales. As an illustration, the CSS scalar field in these coordinates would be φ = f (x) + κτ,

10

(28)

with κ a constant. The generalization to a discrete self-similarity is obvious in these coordinates, and was made in [51]: gµν (τ, xi ) = e−2τ g˜µν (τ, xi ),

where g˜µν (τ, xi ) = g˜µν (τ + ∆, xi ).

(29)

The conformal metric g˜µν does now depend on τ , but only in a periodic manner. Like the continuous symmetry, the discrete version has a geometric formulation [45]: A spacetime is discretely self-similar if there exists a discrete diffeomorphism Φ and a real constant ∆ such that Φ∗ gab = e2∆ gab ,

(30)

where Φ∗ gab is the pull-back of gab under the diffeomorphism Φ. This is our definition of discrete self-similarity (DSS). It can be obtained formally from (21) by integration along ξ over an interval ∆ of the affine parameter. Nevertheless, the definition is independent of any particular vector field ξ. One simple coordinate transformation that brings the Schwarzschild-like coordinates (4) into this form, with the periodicity in τ equivalent to the scaling property (12), was given above in Eqn. (13), as one easily verifies by substitution. The most general ansatz for the massless scalar field compatible with DSS is φ = f (τ, xi ) + κτ,

where

f (τ, xi ) = f (τ + ∆, xi )

(31)

with κ a constant. It should be stressed here that the coordinate systems adapted to CSS (26) or DSS (29) form large classes, even in spherical symmetry. One can fix the surface τ = 0 freely, and can introduce any coordinates xi on it. In particular, in spherical symmetry, τ -surfaces can be chosen to be spacelike, as for example defined by (4) and (13) above, and in this case the coordinate system cannot be global (in the example, t < 0). Alternatively, one can find global coordinate systems, where τ -surfaces must become spacelike at large r, as in the coordinates (34). Moreover, any such coordinate coordinate system can be continuously deformed into one of the same class. As an aside, we mention that self-similarity of the second kind in general relativity was studied by Carter and Henriksen [21] and Coley [37]. The connection with the Newtonian definition is that space and time are rescaled in different ways. To make this a covariant notion one needs a preferred timelike congruence. The 4-velocity ua of a perfect fluid is a natural candidate. The metric gab can then be decomposed into space and time as gab = −ua ub + hab . The homothetic scaling (21) is replaced by Lξ hab = 2hab ,

Lξ ua = Cua ,

(32)

with C 6= 1. This kind of self-similarity has not to date been found in critical collapse. In a possible source of confusion, Evans and Coleman [41] use the term 11

“self-similarity of the second kind”, because they define their self-similar coordinate x as x = r/f (t), with f (t) = tn . Nevertheless, the spacetime they calculate is homothetic, that is, self-similar of the first kind according to the terminology of Carter and Henriksen. The difference is only a coordinate transformation: the t of [41] is not proper time at the origin, but what would be proper time at infinity if the spacetime was truncated at finite radius and matched to an asymptotically flat exterior [40]. There is a large body of research on spherically symmetric self-similar solutions. A detailed review is [19]. Here we should mention only that perfect fluid spherically symmetric self-similar solutions have been examined by Bogoyavlenskii [11], Foglizzo and Henriksen [42], Bicknell and Henriksen [8] and Ori and Piran [79]. Scalar field spherically symmetric CSS solutions were examined by Brady [12]. In these papers, the Einstein equations are reduced to an ODE system by the self-similar spherically symmetric ansatz, which is then discussed as a dynamical system. It is often difficult to regain the spacetime picture from the phase space picture. In particular, it is not clear which solution in these classifications is the critical solution found in perfect fluid collapse simulations, and constructed through a CSS ansatz, by Evans and Coleman [41] (but see [20]). It is also unclear why the scalar field DSS critical solution has κ = 0 in Eqn. (31).

3.2

Gravity regularizes self-similar matter

It is instructive to consider the self-similar solutions of a simple matter field, the massless scalar field, in spherical symmetry without gravity. The general solution of the spherically symmetric wave equation is of course φ(r, t) = r−1 [f (t + r) − g(t − r)] ,

(33)

where f (z) and g(z) are two free functions of one variable ranging from −∞ to ∞. f describes ingoing and g outgoing waves. Regularity at the center r = 0 for all t requires f (z) = g(z) for f (z) a smooth function. Physically this means that ingoing waves move through the center and become outgoing waves. Now we transform to new coordinates x and τ defined by r = e−τ cos x,

t = e−τ sin x,

(34)

and with range −∞ < τ < ∞, −π/2 ≤ x ≤ π/2. These coordinates are adapted to self-similarity, but unlike the x and τ introduced in (13) they cover all of Minkowski space with the exception of the point (t = r = 0). The general solution of the wave equation for t > r can formally be written as φ(r, t) = φ(x, τ )

= −

(tan x + 1)F+ [ln(sin x + cos x) − τ ] (tan x − 1)G+ [ln(sin x − cos x) − τ ] , 12

(35)

through the substitution f (z)/z = F+ (ln z) and g(z)/z = G+ (ln z) for z > 0. Similarly, we define f (z)/z = F− [ln(−z)] and g(z)/z = G− [ln(−z)] for z < 0 to cover the sectors |t| < r and t < −r. Note that F+ (z) and F− (z) together contain the same information as f (z). The condition for regularity at r = 0 for all t > 0 is once more F+ (z) = G+ (z), but we can now also read off that the condition for continuous selfsimilarity φ = φ(x) translates into F+ = const., G+ = const.. Discrete selfsimilarity with scale periodicity ∆, or φ(x, τ ) = φ(x, τ + ∆) translates into F+ (z) = F+ (z + ∆) and G+ (z) = G+ (z + ∆). Any self-similar solution is singular at t = r unless G+ = 0. Similar conclusions are obtained for the sectors |t| < r and t < −r. We conclude that a self-similar solution (continuous or discrete) is either zero everywhere, or else it is regular in at most one of three places: at the center r = 0 for t 6= 0, at the past light cone t = −r, or at the future light cone t = r. (These three cases correspond to F+ = G+ and F− = G− , F+ = F− = 0, and G+ = G− = 0, respectively.) We conjecture that other simple matter fields, such as the perfect fluid, show similar behavior. The presence of gravity changes this singularity structure qualitatively. Dimensional analysis applied to the metric (26) or (29) shows that τ = ∞ [the point (t = r = 0)] is now a curvature singularity (unless the self-similar spacetime is Minkowski). But elsewhere, the solution can be more regular. There is a one-parameter family of exact spherically symmetric scalar field solutions found by Roberts [86] that is regular at both the future and past light cone of the singularity, not only at one of them. (It is singular at the past and future branch of r = 0.) The only solution without gravity with this property is φ = 0. The Roberts solution will be discussed in more detail in section 4.6 below. Similarly, the scale-invariant or scale-periodic solutions found in near-critical collapse simulations are regular at both the past branch of r = 0 and the past light cone (or sound cone, in the case of the perfect fluid). Once more, in the absence of gravity only the trivial solution has this property. I have already argued that the critical solution must be as smooth on the past light cone as elsewhere, as it arises from the collapse of generic smooth initial data. No lowering of differentiability or other unusual behavior should take place before a curvature singularity arises at the center. As Evans first realized, this requirement turns the scale-invariant or scale-periodic ansatz into a boundary value problem between the past branch of r = 0 and the past sound cone, that is, roughly speaking, between x = 0 and x = 1. In the CSS ansatz in spherical symmetry suitable for the perfect fluid, all fields depend only on x, and one obtains an ODE boundary value problem. In a scale-periodic ansatz in spherical symmetry, such as for the scalar field, all fields are periodic in τ , and one obtains a 1+1 dimensional hyperbolic boundary value problem on a coordinate square, with regularity conditions at, say, x = 0 and x = 1, and periodic boundary conditions at τ = 0 and τ = ∆. Well-behaved numerical solutions of these problems have been obtained, with numerical evidence that they are locally unique, and they agree well with the 13

universal solution that emerges in collapse simulations (references are given in the column “Critical solution” of Table 1). It remains an open mathematical problem to prove existence and (local) uniqueness of the solution defined by regularity at the center and the past light cone. One important technical detail should be mentioned here. In the curved solutions, the past light cone of the singularity is not in general r = −t, or x = 1, but is given by x = x0 , or in the case of scale-periodicity, by x = x0 (τ ), with x0 periodic in τ and initially unknown. The same problem arises for the sound cone. It is convenient to make the coordinate transformation 2π x , τ¯ = τ, (36) x ¯= x0 (τ ) ∆ so that the sound cone or light cone is by definition at x ¯ = 1, while the origin is at x ¯ = 0, and so that the period in τ¯ is now always 2π. In the DSS case the periodic function x0 (¯ τ ) and the constant ∆ now appear explicitly in the field equations, and they must be solved for as nonlinear eigenvalues. In the CSS case, the constant x0 appears, and must be solved for as a nonlinear eigenvalue. As an example for a DSS ansatz, we give the equations for the spherically symmetric massless scalar field in the coordinates (13) adapted to self-similarity and in a form ready for posing the boundary value problem. (The equations of [50] have been adapted to the notation of this review.) We introduce the first-order matter variables   √ φ,r φ,t X± = 2πr , (37) ± a α which describe ingoing and outgoing waves. It is also useful to replace α by   ∆ d ln x0 xa D = 1− (38) 2π d¯ τ α as a dependent variable. In the scalar field wave equation (6) we use the Einstein equations (9) and (10) to eliminate a,t and α,r , and obtain ) ( −1   ∂X± ∂X± d ln x0 1 ∆ −1 2 2 2 x ¯ . = (1∓D) (1 − a ) − a X∓ X± − X∓ ± D − ∂x ¯ 2 2π d¯ τ ∂ τ¯ (39) The three Einstein equations (8,9,10) become x ¯ ∂a = a ∂x ¯ x ¯ ∂D = D ∂x ¯ 0 =

1 1 2 2 (1 − a2 ) + a2 (X+ + X− ), 2 2

(40)

2 − a2 ,

(41)

2 2 2 2 (1 − a2 ) + a2 (X+ + X− ) − a2 D−1 (X+ − X− ) −1  2 ∂a d ln x0 ∆ + − . 2π d¯ τ a ∂ τ¯

14

(42)

As suggested by the format of the equations, they can be treated as four evolution equations in x ¯ and one constraint that is propagated by them. The freedom in x0 (¯ τ ) is to be used to make D = 1 at x ¯ = 1. Now x ¯ = 0 and x ¯ = 1 resemble “regular singular points”, if we are prepared to generalize this concept from linear ODEs to nonlinear PDEs. Near x ¯ = 0, the four evolution equations are clearly of the form ∂Z/∂ x ¯ = regular/¯ x. That x ¯ = 1 is also a regular singular ¯ = (1 − D)/(¯ point becomes clearest if we replace D by D x − 1). The “evolution” equation for X+ near x ¯ = 1 then takes the form ∂X+ /∂ x ¯ = regular/(¯ x − 1), while the other three equations are regular. This format of the equations also demonstrates how to restrict from a DSS to a CSS ansatz: one simply drops the τ¯-derivatives. The constraint then becomes algebraic, and the resulting ODE system can be considered to have three rather than four dependent variables. Given that the critical solutions are regular at the past branch of r = 0 and at the past sound cone of the singularity, and that they are self-similar, one would expect them to be singular at the future light cone of the singularity (because after solving the boundary value problem there is no free parameter left in the solution). The real situation is more subtle as we shall see in Section 4.6. As a final remark, it appears that all critical solutions found so far for any matter model, of both type I and type II (see section 4.3 below), do not admit a limit G → 0, so that they are only brought into existence by gravity.

3.3

Universality and scaling

We have seen that the universal solution arising in critical collapse can be constructed semi-analytically from a self-similar ansatz plus regularity conditions. The fact that it is universal up to fine-tuning of one parameter is equivalent to its being an attractor of codimension one. The linearization of that statement around the critical solution is that it has exactly one unstable mode. We now formulate this idea more precisely. For simplicity of notation, we limit ourselves to the spherically symmetric CSS case, for example the perfect fluid. The DSS case is discussed in [51]. Let Z stand for a set of scale-invariant variables of the problem in a first-order formulation. Z(r) is an element of the phase space, and Z(r, t) a solution. The self-similar solution is of the form Z(r, t) = Z∗ (−r/t) = Z∗ (x). [We have chosen the Schwarzschild-like coordinates defined in Eqn. (4), have shifted the origin of t to t = t∗ , and consider only values t < 0.] In the echoing region, where Z∗ dominates, we linearize around it. As the background solution is τ -independent, Z(x, τ ) = Z∗ (x), its linear perturbations can depend on τ only exponentially (with complex exponent λ), that is ∞ X Ci eλi τ fi (x), (43) δZ(x, τ ) = i=1

15

where the Ci are free constants. We can also write this in the more familiar space and time coordinates r and t r = Lxe−τ ,

t = −Le−τ

(44)

already defined in (13) above. To linear order, the solution in the echoing region is then of the form  −λi   ∞  r X t r Ci (p) − + (45) fi − . Z(r, t) ≃ Z∗ − t L t i=1 The coefficients Ci depend in a complicated way on the initial data, and hence on p. If Z∗ is a critical solution, by definition there is exactly one λi with positive real part (in fact it is purely real), say λ1 . As t → 0− , all other perturbations vanish. In the following we consider this limit, and retain only the one growing perturbation. By definition the critical solution corresponds to p = p∗ , so we must have C1 (p∗ ) = 0. Linearizing around p∗ , we obtain −λ1     r  dC r t 1 f1 − . + lim Z(r, t) ≃ Z∗ − (p − p∗ ) − t→0 t dp L t

(46)

This approximate solution explains why the solution Z∗ is universal. It is now also clear why Eqn. (16) holds, that is why we see more of the universal solutions (in the DSS case, more “echos”) as p is tuned closer to p∗ . At an intuitive level, the picture is of either a limit point (in the CSS case), or limit cycle (in the DSS case, as in Fig. 1), in phase space, which is in an attractor in the hypersurface separating black hole from no black hole data. We shall reconsider this picture below in section 5.1. The universal solution is also called the critical solution because it would be revealed up to the singularity τ = ∞ if perfect fine-tuning of p would be possible. A possible source of confusion here is that the critical solution, because it is self-similar, is not asymptotically flat. Nevertheless, it can arise in a region up to finite radius as the limiting case of a family of asymptotically flat solutions. At large radius, it is matched to an asymptotically flat solution which is not universal but depends on the initial data (as does the place of matching.) The following calculation of the critical exponent from the linear perturbations of the critical solution by dimensional analysis was suggested by Evans and Coleman [41] and carried out by Koike, Hara and Adachi [71] and Maison [75]. It was generalized to the discretely self-similar (DSS) case by Gundlach [51]. For simplicity of notation we consider again the CSS case. The solution has the approximate form (46) over a range of t. Now we extract Cauchy data at one particular value of t within that range, namely tp defined by dC1 (47) (p − p∗ )(−tp )−λ1 ≡ ǫ, dp 16

where ǫ is some constant ≪ 1, so that at tp the linear approximation is still valid. (The suffix p indicates that tp depends on p.) At sufficiently small −t, the linear perturbation has grown so much that the linear approximation breaks down. Later on a black hole forms. The crucial point is that we need not follow this evolution in detail, nor does it matter at what amplitude ǫ we consider the perturbation as becoming non-linear. It is sufficient to note that the Cauchy data at t = tp depend on r only in the combination r/tp , namely     r r Z(r, tp ) ≃ Z∗ − + ǫ f1 − . (48) tp tp (tp has of course been defined just so that the coefficient of f1 in this expression is the same for all values of p, namely ǫ.) Furthermore the field equations do not have an intrinsic scale. It follows that the solution based on those data must be exactly [61] of the form   r t , (49) , Z(r, t) = f tp tp

for some function f , throughout, even when the black hole forms and perturbation theory breaks down, and later still after it has settled down and the solution no longer depends on t. (This solution holds only for t > tp , because in its initial data we have neglected the perturbation modes with i > 1, which would be growing, not decaying, towards the past.) Because the black hole mass has dimension length, it must be proportional to tp , the only length scale in the solution, 1 M ∝ tp ∝ (p − p∗ ) λ1 , (50)

and we have found the critical exponent γ = 1/λ1 . When the critical solution is DSS, the scaling law is modified. This was predicted by [51] and predicted independently and verified in collapse simulations by Hod and Piran [64]. On the straight line relating ln M to ln(p − p∗ ), a periodic “wiggle” or “fine structure” of small amplitude is superimposed: ln M = γ ln(p − p∗ ) + c + f [γ ln(p − p∗ ) + c],

(51)

with f (z) = f (z + ∆). The periodic function f is again universal with respect to families of initial data, and there is only one parameter c that depends on the family of initial data, corresponding to a shift of the wiggly line in the ln(p − p∗ ) direction. (No separate adjustment in the ln M direction is possible.)

4 4.1

Extensions of the basic scenario More matter models

Choptuik’s results have been confirmed for a variety of other matter models. In some of these, qualitatively new phenomena were discovered, and we review this 17

Table 1: An overview of numerical work in critical collapse. Question marks denote missing links. Matter model Collapse Critical Perturbations simulations solution Perfect fluid – k = 1/3 [41] [41] [71] – general k ? [75] [75] Real scalar field – massless, min. coupled [24, 25, 26] [50] [51] [26, 14] [59, 55]1 [59, 55]1 – massive – conformally coupled [25] ? ? 2-d sigma model – complex scalar (κ = 0) [27] [51]2 , [60]3 [51]2 , [61]3 – axion-dilaton (κ = 1) [57] [39, 57] [57] – scalar-Brans-Dicke (κ > 0) [74] – general κ including κ < 0 ? [62] [62] Scalar electrodynamics [63] [55]4 [55]4 SU (2) Yang-Mills [29] [52] [52] SU (2) Skyrme model [10] [10] [10] Axisymmetric vacuum [1, 2] ? ? body of work by phenomena rather than chronologically or by matter models. A presentation by matter models is given in Table 1 for completeness. An exceptional case is spherically symmetric dust collapse. Here, the entire spacetime, the Tolman-Bondi solution, is given in closed form from the initial velocity and density profiles. Excluding shell crossing singularities, there is a “phase transition” between initial data forming naked singularities at the center and data forming black holes. Which of the two happens depends only the leading terms in an expansion of the initial data around r = 0 [31, 70]. One could argue that this fact also makes the matter model rather unphysical.

4.2

CSS and DSS critical solutions

As we have seen, a critical solution is one that sits on the boundary of black hole formation, and has exactly one “growing mode”, so that it acts as an intermediate attractor (Evans). All one-parameter families of initial data crossing 1 The critical solution and its perturbations for the massive scalar field are asymptotic to those of the massless scalar. 2 The (DSS) critical solution for the real massless scalar field is also the critical solution for the complex scalar field. The additional perturbations are all stable [51]. 3 There is also a CSS solution [60], but it has three unstable modes, not only one [61]. 4 The scalar electrodynamics critical solution is again the real scalar field critical solution. Its perturbations are those of the complex scalar field.

18

that boundary are then “funnelled” (Eardley) through that one solution. So far, we have seen an example each of a critical solution with discrete and with continuous self-similarity. There may be regular CSS or DSS solutions with more than one growing mode, but they will not appear in Choptuik type fine-tuning. An example for this is provided by the spherically symmetric massless complex scalar field. Hirschmann and Eardley [60] found a way of constructing a CSS scalar field solution by making the scalar field φ complex but limiting it to the ansatz φ = eiωτ f (x), (52) with ω a real constant and f real. The metric is then homothetic, while the scalar field shows a trivial kind of “echoing” in the complex phase. Later, they found that this solution has three modes with Reλ > 0 [61] and is therefore not the critical solution. Gundlach [51] examined complex scalar field perturbations around Choptuik’s real scalar field critical solution and found that only one of them, purely real, has Reλ > 0, so that the real scalar field critical solution is a critical solution (up to an overall complex phase) also for the free complex scalar field. This had been seen already in collapse calculations [27]. As the symmetry of the critical solution, CSS or DSS, depends on the matter model, it is interesting to investigate critical behavior in parameterized families of matter models. Two such one-parameter families have been investigated. The first one is the spherical perfect fluid with equation of state p = kρ for arbitrary k. Maison [75] constructed the regular CSS solutions and its linear perturbations for a large number of values of k. In each case, he found exactly one growing mode, and was therefore able to predict the critical exponent. (To my knowledge, these critical exponents have not yet been verified in collapse simulations.) As Ori and Piran before [79], he found that there are no regular CSS solutions for k > 0.88. There is nothing in the equation of state to explain this. In particular, the perfect fluid is well behaved up to k < 1. It remains unknown what happens in critical collapse for k > 0.88 Black hole formation may begin with a minimum mass. (In the absence of a mass scale in the field equations, this mass gap would depend on the family.) Alternatively, there may be a DSS critical solution. The fact that the k = 1 perfect fluid is equivalent to a massless scalar field, which does have a DSS critical solution, hints in this direction. Nevertheless, a scalar field solution corresponds to a perfect fluid solution only if φ,a is everywhere timelike, and this is not true for Choptuik’s universal solution. The second one-parameter family of matter models was suggested by Hirschmann and Eardley [62], who looked for a natural way of introducing a non-linear self-interaction for the (complex) scalar field without introducing a scale. (We discuss dimensionful coupling constants in the following sections.) They investigated the model described by the action   Z 2|∇φ|2 √ . (53) g R− S= (1 − κ|φ|2 )2 19

Note that φ is now complex, and the parameter κ is real and dimensionless. This is a 2-dimensional sigma model with a target space metric of constant curvature (namely κ), minimally coupled to gravity. Moreover, for κ > 0 there are (nontrivial) field redefinitions which make this model equivalent to a real massless scalar field minimally coupled to Brans-Dicke gravity, with the BransDicke coupling given by 1 3 . (54) ωBD = − + 2 8κ In particular, κ = 1 (ωBD = −11/8) corresponds to an axion-dilaton system arising in string theory [39]. κ = 0 is the free complex scalar field coupled to Einstein gravity). Hirschmann and Eardley calculated a CSS solution and its perturbations and concluded that it is the critical solution for κ > 0.0754, but has three unstable modes for κ < 0.0754. For κ < −0.28, it acquires even more unstable modes. The positions of the mode frequencies λ in the complex plane vary continuously with κ, and these are just values of κ where a complex conjugate pair of frequencies crosses the real axis. The results of Hirschmann and Eardley confirm and subsume collapse simulation results by Liebling and Choptuik [74] for the scalar-Brans-Dicke system, and collapse and perturbative results on the axion-dilaton system by Hamad´e, Horne and Stewart [57]. Where the CSS solution fails to be the critical solution, a DSS solution takes over. In particular, for κ = 0, the free complex scalar field, the critical solution is just the real scalar field DSS solution of Choptuik.

4.3

Black hole thresholds with a mass gap

The first models in which critical phenomena were observed did not have any length scales in the field equations. Later, models were examined which have one such scale. Collapse simulations were carried out for the spherically symmetric SU (2) Einstein-Yang-Mills system by Choptuik, Chmaj and Bizon [29]. In finetuning one-parameter families of data to the black-hole threshold, they found two different kinds of critical phenomena, dominated by two different critical solutions. Which kind of behavior arises appears to depend on the qualitative shape of the initial data. In one kind of behavior, black hole formation turns on at an infinitesimal mass with the familiar power-law scaling, dominated by a DSS critical solution. In the other kind, black hole formation turns on at a finite mass, and the critical solution is now a static, asymptotically flat solution which had been found before by Bartnik and McKinnon [7]. It was also known before that this solution (the least massive one of a discrete family) had exactly two unstable perturbation modes [73]. The ansatz of Choptuik, Chmaj and Bizon further allowed for only one of these unstable modes, with one sign of these leading to collapse and the other to dispersion of the solution. The Bartnik-McKinnon solution is then a critical solution within this ansatz, in the sense of being an attractor of codimension one on the black hole threshold. Choptuik, Chmaj and Bizon labelled the two kinds of critical behavior type II and type I respectively, 20

corresponding to a second- and a first-order phase transition. The newly found, type I critical phenomena show a scaling law that is mathematically similar to the black hole mass scaling observed in type II critical phenomena. Let ∂/∂t be the static Killing vector of the critical solution. Then the perturbed critical solution is of the form Z(r, t) = Z∗ (r) +

dC1 (p − p∗ )eλ1 t f1 (r) + decaying modes. dp

(55)

This is similar to Eqn. (46), but the growth of the unstable mode is now exponential in t, not in ln t. We again define a time tp by dC1 (p − p∗ )eλ1 tp ≡ ǫ, dp

(56)

but now the initial data at tp are Z(r, tp ) ≃ Z∗ (r) + ǫ f1 (r) ,

(57)

so that that the final black hole mass is independent of p − p∗ . (It is of the order of the mass of the static critical solution.) The scaling is only apparent in the lifetime of the critical solution, which we can take to be tp . It is tp = −

1 ln(p − p∗ ) + const. λ1

(58)

The type I critical solution can also have a discrete symmetry, that is, can be periodic in time instead of being static. This behavior was found in collapse situations of the massive scalar field by Brady, Chambers and Gon¸calves [14]. Previously, Seidel and Suen [87] had constructed periodic, asymptotically flat, spherically symmetric self-gravitating massive scalar field solutions they called oscillating soliton stars. By dimensional analysis, the scalar field mass m sets an overall scale of 1/m (in units G = c = 1). For given m, Seidel and Suen found a one-parameter family of such solutions with two branches. The more compact solution for a given ADM mass is unstable, while the more extended one is stable to spherical perturbations. Brady, Chambers and Gon¸calves (BCG) report that the type I critical solutions they find are from the unstable branch of the Seidel and Suen solutions. Therefore we are seeing a one-parameter family of (type I) critical solutions, rather than an isolated critical solution. BCG in fact report that the black hole mass gap does depend on the initial data. They find a small wiggle in the mass of the critical solution which is periodic in ln(p − p∗ ), and which should have the same explanation [51] as that found in the mass of the black hole in type II DSS critical behavior. If type I or type II behavior is seen appears to depend mainly on the ratio of the length scale of the initial data to the length scale 1/m. One point in the results of BCG is worth expanding on. In the critical phenomena that were first observed, with an isolated critical solution, only one 21

number’s worth of information, namely the separation p − p∗ of the initial data from the black hole threshold, survives to the late stages of the time evolution. This is true for both type I and type II critical phenomena. In type II phenomena, p − p∗ determines the black hole mass, while in both type I and II it also determines the lifetime of the critical solution (the number of echos). Recall that our definition of a critical solution is one that has exactly one unstable perturbation mode, with a black hole formed for one sign of the unstable mode, but not for the other. This definition does not exclude an n-dimensional family of critical solutions. Each solution in the family would then have n marginal modes leading to neighboring critical solutions, as well as the one unstable mode. n+1 numbers’ worth of information would survive from the initial data, and the mass gap in type I, or the critical exponent for the black hole mass in type II, for example, would depend on the initial data through n parameters. In other words, universality would exist in diminished form. The results of BCG are an example of a one-parameter family of type I critical solutions. Recently, Brodbeck et al. [16] have shown, under the assumption of linearization stability, that there is a one-parameter family of stationary, rotating solutions beginning at the (spherically symmetric) Bartnik-McKinnon solution. This could turn out to be a second one-parameter family of type I critical solutions, provided that the Bartnik-McKinnon solution does not have any unstable modes outside spherical symmetry (which has not yet been investigated) [85]. Bizo´ n and Chmaj have studied type I critical collapse of an SU (2) Skyrme model coupled to gravity, which in spherical symmetry with a hedgehog ansatz is characterized by one field F (r, t) and one dimensionless coupling constant α. Initial data F (r) ∼ tanh(r/p), F˙ (r) = 0 surprisingly form black holes for both large and small values of the parameter p, while for an intermediate range of p the endpoint is a stable static solution called a skyrmion. (If F was a scalar field, one would expect only one critical point on this family.) The ultimate reason for this behavior is the presence of a conserved integer “baryon number” in the matter model. Both phase transitions along this one-parameter family are dominated by a type I critical solution, that is a different skyrmion which has one unstable mode. In particular, an intermediate time regime of critical collapse evolutions agrees well with an ansatz of the form (55), where Z∗ , f1 and λ were obtained independently. It is interesting to note that the type I critical solution is singular in the limit α → 0, which is equivalent to G → 0, because the known type II critical solutions for any matter model also do not have a weak gravity limit. Apparently, type I critical phenomena can arise even without the presence of a scale in the field equations. A family of exact spherically symmetric, static, asymptotically flat solutions of vacuum Brans-Dicke gravity given by van Putten was found by Choptuik, Hirschmann and Liebling [30] to sit at the black hole-threshold and to have exactly one growing mode. This family has two parameters, one of which is an arbitrary overall scale.

22

4.4

Approximate self-similarity and universality classes

As we have seen, the presence of a length scale in the field equations can give rise to static (or oscillating) asymptotically flat critical solutions and a mass gap at the black hole threshold. Depending on the initial data, this scale can also become asymptotically irrelevant as a self-similar solution reaches ever smaller spacetime scales. This behavior was already noticed by Choptuik in the collapse of a massive scalar field, or one with a potential term generally [26] and confirmed by Brady, Chambers and Gon¸calves [14]. It was also seen in the spherically symmetric EYM system [29]. In order to capture the notion of an asymptotically self-similar solution, one may set the arbitrary scale L in the definition (13) of τ to the scale of the field equations, here 1/m. Introducing suitable dimensionless first-order variables Z (such as a, α, φ, rφ,r and rφ,t for the spherically symmetric scalar field), one can write the field equations as a first order system
(59) F Z, Z,x , Z,τ , e−τ = 0.

Every appearance of m gives rise to an appearance of e−τ . If the field equations contain only positive integer powers of m, one can make an ansatz for the critical solution of the form ∞ X e−nτ Zn (x) (60) Z∗ (x, τ ) = n=0

where each Zn (x) is calculated recursively from the preceding ones. For large enough τ (on spacetime scales small enough, close enough to the singularity), this infinite series is expected to converge. A similar ansatz can be made for the linear perturbations of Z∗ , and solved again recursively. Fortunately, one can calculate the leading order background term Z0 on its own, and obtain the exact echoing period ∆ in the process (in the case of DSS). Similarly, one can calculate the leading order perturbation term on the basis of Z0 alone, and obtain the exact value of the critical exponent γ in the process. This procedure was carried out by Gundlach [52] for the Einstein-Yang-Mills system, and by Gundlach and Mart´ın-Garc´ıa [55] for massless scalar electrodynamics. Both systems have a single scale 1/e (in units c = G = 1), where e is the gauge coupling constant. The leading order term Z0 in the expansion of the self-similar critical solution Z∗ obeys the equation F (Z0 , Z0,x , Z0,τ , 0) = 0. (61) Clearly, the critical solution is independent of the overall scale L0 . By a similar argument, so are its perturbations, and therefore the critical exponent γ. Therefore, all systems with a single length scale L0 in the field equations are in one universality class [59, 55]. The massive scalar field, for any value of m, or massless scalar electrodynamics, for any value of e, are in the same universality class as the massless scalar field. This notion of universality classes is fundamentally the same as in statistical mechanics. 23

If there are several scales L0 , L1 , L2 etc. present in the problem, a possible approach is to set the arbitrary scale in (13) equal to one of them, say L0 , and define the dimensionless constants li = Li /L0 from the others. The size of the universality classes depends on where the li appear in the field equations. If a particular Li appears in the field equations only in positive integer powers, the corresponding li appears only multiplied by e−τ , and will be irrelevant in the scaling limit. All values of this li therefore belong to the same universality class. As an example, adding a quartic self-interaction λφ4 to the massive scalar field, for example, gives rise to the dimensionless number λ/m2 , but its value is an irrelevant (in the language of renormalisation group theory) parameter. All self-interacting scalar fields are in fact in the same universality class. Contrary to the statement in [55], I would now conjecture that massive scalar electrodynamics, for any values of e and m, forms a single universality class in type II critical phenomena. Examples of dimensionless parameters which do change the universality class are the k of the perfect fluid, the κ of the 2-dimensional sigma model, or a conformal coupling of the scalar field.

4.5

Beyond spherical symmetry

Every aspect of the basic scenario: CSS and DSS, universality and scaling applies directly to a critical solution that is not spherically symmetric, but all the models we have described are spherically symmetric. There are only two exceptions to date: a numerical investigation of critical collapse in axisymmetric pure gravity [1], and a study of the nonspherical perturbations the perfect fluid critical solution [53]. They correspond to two related questions in going beyond spherical symmetry. Are there critical phenomena in gravitational collapse far from spherical symmetry? And: are the critical phenomena in the known spherically symmetric examples destroyed by small deviations from spherical symmetry? 4.5.1

Axisymmetric gravitational waves

The paper of Abrahams and Evans [1] was the first paper on critical collapse to be published after Choptuik’s PRL, but it remains the only one to investigate a non-spherically symmetric situation, and therefore also the only one to investigate critical phenomena in the collapse of gravitational waves in vacuum. Because of its importance, we summarize its contents here with some technical detail. The physical situation under consideration is axisymmetric vacuum gravity. The numerical scheme uses a 3+1 split of the spacetime. The ansatz for the spacetime metric is i h ds2 = −α2 dt2 +φ4 e2η/3 (dr + β r dt)2 + r2 e2η/3 (dθ + β θ dt)2 + e−4η/3 r2 sin2 θ dϕ2 , (62)

24

parameterized by the lapse α, shift components β r and β θ , and two independent coefficients φ and η in the 3-metric. All are functions of r, t and θ. The fact that dr2 and r2 dθ2 are multiplied by the same coefficient is called quasiisotropic spatial gauge. The variables for a first-order-in-time version of the Einstein equations are completed by the three independent components of the extrinsic curvature, Kθr , Krr , and Kϕϕ . In order to obtain initial data obeying the constraints, η and Kθr are given as free data, while the remaining components of the initial data, namely φ, Krr , and Kϕϕ , are determined by solving the Hamiltonian constraint and the two independent components of the momentum constraint respectively. There are five initial data variables, and three gauge variables. Four of the five initial data variables, namely η, Kθr , Krr , and Kϕϕ , are updated from one time step to the next via evolution equations. As many variables as possible, namely φ and the three gauge variables α, β r and β θ , are obtained at each new time step by solving elliptic equations. These elliptic equations are the Hamiltonian constraint for φ, the gauge condition of maximal slicing (Ki i = 0) for α, and the gauge conditions gθθ = r2 grr and grθ = 0 for β r and β θ (quasi-isotropic gauge). For definiteness, the two free functions, η and Kθr , in the initial data were chosen to have the same functional form they would have in a linearized gravitational wave with pure (l = 2, m = 0) angular dependence. Of course, depending on the overall amplitude of η and Kθr , the other functions in the initial data will deviate more or less from their linearized values, as the non-linear initial value problem is solved exactly. In axisymmetry, only one of the two degrees of freedom of gravitational waves exists. In order to keep their numerical grid as small as possible, Abrahams and Evans chose the pseudo-linear waves to be purely ingoing. (In nonlinear general relativity, no exact notion of ingoing and outgoing waves exists, but this ansatz means that the wave is initially ingoing in the low-amplitude limit.) This ansatz (pseudo-linear, ingoing, l = 2), reduced the freedom in the initial data to one free function of advanced time, I (2) (v). A suitably peaked function was chosen. Limited numerical resolution (numerical grids are now two-dimensional, not one-dimensional as in spherical symmetry) allowed Abrahams and Evans to find black holes with masses only down to 0.2 of the ADM mass. Even this far from criticality, they found power-law scaling of the black hole mass, with a critical exponent γ ≃ 0.36. Determining the black hole mass is not trivial, and was done from the apparent horizon surface area, and the frequencies of the lowest quasinormal modes of the black hole. There was tentative evidence for scale echoing in the time evolution, with ∆ ≃ 0.6, with about three echos seen. This corresponds to a scale range of about one order of magnitude. By a lucky coincidence, ∆ is much smaller than in all other examples, so that several echos could be seen without adaptive mesh refinement. The paper states that the function η has the echoing property η(e∆ r, e∆ t) = η(r, t). If the spacetime is DSS in the sense defined above, the same echoing property is expected to hold also for α, φ, β r and r−1 β θ , as one sees by applying the coordinate transformation (13) to (62). 25

In a subsequent paper [2], universality of the critical solution, echoing period and critical exponent was demonstrated through the evolution of a second family of initial data, one in which η = 0 at the initial time. In this family, black hole masses down to 0.06 of the ADM mass were achieved. Further work on critical collapse far away from spherical symmetry would be desirable, but appears to be held up by numerical difficulty. 4.5.2

Perturbing around sphericity

A different, and technically simpler, approach is to take a known critical solution in spherical symmetry, and perturb it using nonspherical perturbations. Addressing this perturbative question, Gundlach [53] has studied the generic non-spherical perturbations around the critical solution found by Evans and Coleman [41] for the p = 13 ρ perfect fluid in spherical symmetry. He finds that there is exactly one spherical perturbation mode that grows towards the singularity (confirming the previous results [71, 75]). He finds no growing nonspherical modes at all. The main significance of this result, even though it is only perturbative, is to establish one critical solution that really has only one unstable perturbation mode within the full phase space. As the critical solution itself has a naked singularity (see Section 4.6), this means that there is, for this matter model, a set of initial data of codimension one in the full phase space of general relativity that forms a naked singularity. In hindsight, this result also fully justifies the attention that critical phenomena gravitational collapse have won as a “natural” route to naked singularities.

4.6

Critical phenomena and naked singularities

Choptuik’s result have an obvious bearing on the issue of cosmic censorship. (For a general review of cosmic censorship, see [93].) As we shall see in this section, the critical spacetime has a naked singularity. This spacetime can be approximated arbitrarily well up to fine-tuning of a generic parameter. A region of arbitrarily high curvature is seen from infinity as fine-tuning is improved. Critical collapse provides a set of smooth initial data from which a naked singularity is formed. In spite of news to the contrary, it violates neither the letter nor the spirit of cosmic censorship because this set is of measure zero. Nevertheless it comes closer than would have been imagined possible before the work of Choptuik. First of all, the set of data is of codimension one, certainly in the space of spherical asymptotically flat data, and apparently [53] also in the space of all asymptotically flat data. This means that one can fine-tune any generic parameter, whichever comes to hand, as long as it parameterizes a smooth curve in the space of initial data. Secondly, critical phenomena seem to be generic with respect to matter models, including realistic matter models with intrinsic scales. For a hypothetical experiment to create a Planck-sized black hole

26

in the laboratory through a strong explosion, this would mean that one could fine-tune any one design parameter of the bomb to the black hole threshold, without requiring much control over its detailed effects on the explosion. The metric of the critical spacetime is of the form e−2τ times a regular metric. From this general form alone, one can conclude that τ = ∞ is a curvature singularity, where Riemann and Ricci invariants blow up like e4τ , and which is at finite proper time from regular points. The Weyl tensor with index position C a bcd is conformally invariant, so that components with this index position remain finite as τ → ∞. In this property it resembles the initial singularity in Penrose’s Weyl tensor conjecture rather than the final singularity in generic gravitational collapse. This type of singularity is called “conformally compactifiable” [90] or “isotropic” [48]. Is the singularity naked, and is it timelike, null or a “point”? The answer to these questions remains confused, partly because of coordinate complications, partly because of the difficulty of investigating the singular behavior of solutions numerically. Choptuik’s, and Evans and Coleman’s, numerical codes were limited to the region t < 0, in the Schwarzschild-like coordinates (4), with the origin of t adjusted so that the singularity is at t = 0. Evans and Coleman conjectured that the singularity is shrouded in an infinite redshift based on the fact that α grows as a small power of r at constant t. This is directly related to the fact that a goes to a constant a∞ > 1 as r → ∞ at constant t, as one can see from the Einstein equation (9). This in turn means simply that the critical spacetime is not asymptotically flat, but asymptotically conical at spacelike infinity, with the Hawking mass proportional to r. Hamad´e and Stewart [58] evolved near-critical scalar field spacetimes on a double null grid, which allowed them to follow the time evolution up to close to the future light cone of the singularity. They found evidence that this light cone is not preceded by an apparent horizon, that it is not itself a (null) curvature singularity, and that there is only a finite redshift along outgoing null geodesics slightly preceding it. (All spherically symmetric critical spacetimes appear to be qualitatively alike as far as the singularity structure is concerned, so that what we say about one is likely to hold for the others.) Hirschmann and Eardley [60] were the first to continue a critical solution itself right up to the future light cone. They examined a CSS complex scalar field solution that they had constructed as a nonlinear ODE boundary value problem, as discussed in Section 3.2. (This particular one is not a proper critical solution, but that should not matter for the global structure.) They continued the ODE evolution in the self-similar coordinate x through the coordinate singularity at t = 0 up to the future light cone by introducing a new self-similarity coordinate x. The self-similar ansatz reduces the field equations to an ODE system. The past and future light cones are regular singular points of the system, at x = x1 and x = x2 . At these “points” one of the two independent solutions is regular and one singular. The boundary value problem that originally defines the critical solution corresponds to completely suppressing the singular solution at x = x1 27

(the past light cone). The solution can be continued through this point up to x = x2 . There it is a mixture of the regular and the singular solution. We now state this more mathematically. The ansatz of Hirschmann and Eardley for the self-similar complex scalar field is (we slightly adapt their notation) φ(x, τ ) = f (x)eiωτ , a = a(x), α = α(x), (63) with ω a real constant. Near the future light cone they find that f is approximately of the form f (x) ≃ Creg (x) + (x − x2 )(iω+1)(1+ǫ) Csing (x),

(64)

with Creg (x) and Csing(x) regular at x = x2 , and ǫ a small positive constant. The singular part of the scalar field oscillates an infinite number of times as x → x2 , but with decaying amplitude. This means that the scalar field φ is just differentiable, and that therefore the stress tensor is just continuous. It is crucial that spacetime is not flat, or else ǫ would vanish. For this in turn it is crucial that the regular part Creg of the solution does not vanish, as one sees from the field equations. The only other case in which the critical solution has been continued up to the future light cone is Choptuik’s real scalar field solution [51]. Let X+ and X− be the ingoing and outgoing wave degrees of freedom respectively defined in (37). At the future light cone x = x2 the solution has the form X− (x, τ ) X+ (x, τ )

≃

≃

f− (x, τ ),

(65) ǫ

f+ (x, τ ) + (x − x2 ) fsing (x, τ − C ln x),

(66)

where C is a positive real constant, f− , f+ and fsing are regular real functions with period ∆ in their second argument, and ǫ is a small positive real constant. (We have again simplified the original notation.) Again, the singular part of the solution oscillates an infinite number of times but with decaying amplitude. Gundlach concludes that the scalar field, the metric coefficients, all their first derivatives, and the Riemann tensor exist, but that is as far as differentiability goes. (Not all second derivatives of the metric exist, but enough to construct the Riemann tensor.) If either of the regular parts f− or f+ vanished, spacetime would be flat, ǫ would vanish, and the scalar field itself would be singular. In this sense, gravity regularizes the self-similar matter field ansatz. In the critical solution, it does this perfectly at the past lightcone, but only partly at the future lightcone. Perhaps significantly, spacetime is almost flat at the future horizon in both the examples, in the sense that the Hawking mass divided by r is a very small number, as small as 10−6 (but not zero according to numerical work by Horne [66]) in the spacetime of Hirschmann and Eardley. In summary, the future light cone (or Cauchy horizon) of these two critical spacetimes is not a curvature singularity, but it is singular in the sense that differentiability is lower than elsewhere in the solution. Locally, one can continue 28

the solution through the future light cone to an almost flat spacetime (the solution is of course not unique). It is not clear, however, if such a continuation can have a regular center r = 0 (for t > 0), although this seems to have been assumed by some authors. A priori, one should expect a conical singularity, with a (small) defect angle at r = 0. The results just discussed were hampered by the fact that they are investigations of singular spacetimes that are only known in numerical form, with a limited precision. As an exact toy model we consider an exact spherically symmetric, CSS solution for massless real scalar field that was apparently first discovered by Roberts [86] and then re-discovered in the context of critical collapse by Brady [13] and Oshiro et al. [80]. We use the notation of Oshiro et al. The solution can be given in double null coordinates as ds2

=

r2 (u, v) = φ(u, v) =

−du dv + r2 (u, v) dΩ2 ,  1 (1 − p2 )v 2 − 2vu + u2 , 4 1 (1 − p)v − u ln , 2 (1 + p)v − u

(67) (68) (69)

with p a constant parameter. (Units G = c = 1.) Two important curvature indicators, the Ricci scalar and the Hawking mass, are R=

p2 uv , 2r4

M =−

p2 uv . 8r

(70)

The center r = 0 has two branches, u = (1 + p)v in the past of u = v = 0, and u = (1 − p)v in the future. For 0 < p < 1 these are timelike curvature singularities. The singularities have negative mass, and the Hawking mass is negative in the past and future light cones. One can cut these regions out and replace them by Minkowski space, not smoothly of course, but without creating a δ-function in the stress-energy tensor. The resulting spacetime resembles the critical spacetimes arising in gravitational collapse in some respects: it is selfsimilar, has a regular center r = 0 at the past of the curvature singularity u = v = 0 and is continuous at the past light cone. It is also continuous at the future light cone, and the future branch of r = 0 is again regular. It is interesting to compare this with the genuine critical solutions that arise as attractors in critical collapse. They are as regular as the Roberts solution (analytic) at the past r = 0, more regular (analytic versus continuous) at the past light cone, as regular (continuous) at the future light cone and, it is to be feared, less regular at the future branch of r = 0: In contrary to previous claims [60, 49] there may be no continuation through the future sound or light cone that does not have a conical singularity at the future branch of r = 0. The global structure still needs to be clarified for all known critical solutions. In summary, the critical spacetimes that arise asymptotically in the finetuning of gravitational collapse to the black-hole threshold have a curvature 29

singularity that is visible at infinity with a finite redshift. The Cauchy horizon of the singularity is mildly singular (low differentiability), but the curvature is finite there. It is unclear at present if the singularity is timelike or if there exists a continuation beyond the Cauchy horizon with a regular center, so that the singularity is limited, loosely speaking, to a point. Further work should be able to clarify this. In any case, the singularity is naked and the critical solutions therefore provide counter-examples to some formulations of cosmic censorship which state that naked singularities cannot arise from smooth initial data in reasonable matter models. It is now clear that one must refine this to state that there is no open ball of smooth initial for naked singularities. Recent analytic work by Christodoulou also comes to this conclusion [36].

4.7

Black hole charge and angular momentum

Given the scaling power law for the black hole mass in critical collapse, one would like to know what happens if one takes a generic one-parameter family of initial data with both electric charge and angular momentum (for suitable matter), and fine-tunes the parameter p to the black hole threshold. Does the mass still show power-law scaling? What happens to the dimensionless ratios L/M 2 and Q/M , with L the black hole angular momentum and Q its electric charge? Tentative answers to both questions have been given using perturbations around spherically symmetric uncharged collapse. 4.7.1

Charge

Gundlach and Mart´ın-Garc´ıa [55] have studied scalar massless electrodynamics in spherical symmetry. Clearly, the real scalar field critical solution of Choptuik is a solution of this system too. Less obviously, it remains a critical solution within massless (and in fact, massive) scalar electrodynamics in the sense that it still has only one growing perturbation mode within the enlarged solution space. Some of its perturbations carry electric charge, but as they are all decaying, electric charge is a subdominant effect. The charge of the black hole in the critical limit is dominated by the most slowly decaying of the charged modes. From this analysis, a universal power-law scaling of the black hole charge Q ∼ (p − p∗ )δ

(71)

was predicted. The predicted value δ ≃ 0.88 of the critical exponent (in scalar electrodynamics) was subsequently verified in collapse simulations by Hod and Piran [63]. (The mass scales with γ ≃ 0.37 as for the uncharged scalar field.) General considerations using dimensional analysis led Gundlach and Mart´ınGarc´ıa to the general prediction that the two critical exponents are always related, for any matter model, by the inequality δ ≥ 2γ. 30

(72)

This has not yet been verified in any other matter model. 4.7.2

Angular momentum

Gundlach’s results on non-spherically symmetric perturbations around spherical critical collapse of a perfect fluid [53] allow for initial data, and therefore black holes, with infinitesimal angular momentum. All nonspherical perturbations decrease towards the singularity. The situation is therefore similar to scalar electrodynamics versus the real scalar field. The critical solution of the more special model (here, the strictly spherically symmetric fluid) is still a critical solution within the more general model (a slightly nonspherical and slowly rotating fluid). In particular, axial perturbations (also called odd-parity perturbations) with angular dependence l = 1 will determine the angular momentum of the black hole produced in slightly supercritical collapse. Using a perturbation analysis similar to that of Gundlach and Mart´ın-Garc´ıa [55], Gundlach [54] has derived the angular momentum scaling i h ~ = Re (A ~ + iB)(p ~ (73) L − p∗ )µ+iω ,

~ and B ~ are family-dependent constants, and the complex critical expowhere A nent µ + iω is universal. For p = ρ/3, he predicts the values of µ and ω. In the special of axisymmetry, this result reduces to L = Lz = (p − p∗ )µ A cos[ω ln(p − p∗ ) + c],

(74)

which is rather surprising. The explanation is of course that near the black hole threshold the initial data, including the initial angular momentum, are totally forgotten, while the oscillating angular momentum of the black hole is a subdominant effect. These results have not yet been tested against numerical collapse simulations. Traschen [91] has drawn attention to a different possible connection between critical phenomena and black hole charge. Consider the equation of motion for a massive charged scalar test field on a fixed charged black hole background. (The test field is coupled via ∇ + ieA, but its back-reaction on both the metric and the Maxwell field is neglected.) In the limit Q = M for the background black hole, and near the horizon this linear equation has a scale-invariance not present in non-extremal black holes because the surface gravity, which otherwise sets a scale, vanishes. The equation therefore admits self-similar solutions. Traschen suggests these are DSS, but there is no argument why either a CSS or a DSS solution should play a special role.

5

More speculative aspects

31

5.1

The renormalisation group as a time evolution

It has been pointed out by Argyres [3], Koike, Hara and Adachi [71] and others that the time evolution near the critical solution can be considered as a renormalisation group flow on the space of initial data. The calculation of the critical exponent in section 3.3 is in fact mathematically identical with that of the critical exponent governing the correlation length near the critical point in statistical mechanics [96], if one identifies the time evolution in the time coordinate τ and spatial coordinate x with the renormalisation group flow. For simple parabolic or hyperbolic differential equations, a discrete renormalisation (semi)group acting on their solutions has been defined in the following way [47]. Evolve initial data over a certain finite time interval, then rescale the final data in a certain way. Solutions which are fixed points under this transformation are scale-invariant, and may be attractors. In the context of the spherically symmetric scalar field described in section 2.1 this prescription takes the following form. Take free data φ0 (r) = φ(t0 , r), Π0 (r) = Π(t0 , r). Evolve them from time t0 to time t1 = e−∆ (t0 − t∗ ) + t∗ . Obtain new data φ1 (r) = φ(t1 , e−∆ r) and Π1 (r) = e−∆ Π(t1 , e−∆ r). One can introduce new coordinates and fields such that the renormalisation transformation becomes a simple time evolution without any explicit rescaling. For the scalar field model, this form of the transformation is simply Z0 (x) = Z(τ0 , x) to Z1 (x) = Z(τ0 + ∆, x), where Z stands for the fields φ and rΠ. The coordinates x and τ replace r and t. While this approach looks promising, its application to general relativity has not yet been achieved. In general relativity as in other field theories or in dynamical systems, a solution is determined (at least locally) by an initial data set. (In general relativity, a solution is a spacetime, and the initial data are the first and second fundamental forms of a spacelike hypersurface, plus suitable matter variables.) A crucial distinctive feature of general relativity, however, is that a solution does not correspond to a unique trajectory in the space of initial data. This is because a spacetime can be sliced in different ways, and on each slice one can have different coordinate systems. Infinitesimally, this slicing and coordinate freedom is parameterized by the lapse and shift. They can be set freely, independently of the initial data, and they influence only the coordinates on the spacetime, not the spacetime itself. What coordinates should one use then when describing a time evolution in GR as a renormalisation group transformation on the space of initial data? For a given self-similar spacetime, there are preferred coordinates adapted to the symmetry and defined by Eqn. (29). These are far from unique. Nevertheless, one can choose one of them, and then extend this choice of coordinates to linear perturbations around the self-similar solution. In this linear regime, one really obtains a time evolution such that a phase space diagram of the type of Fig. 1 makes sense: On the limit cycle, the same Cauchy data, expressed in the same space coordinates, return periodically. (The overall scale decreases

32

in each period, and is suppressed in Fig. 1.) Hara, Koike and Adachi [59] have used these coordinates in numerical work in order to find the spectrum of perturbations of the critical solution by evolving a generic perturbation in time, and peeling off the individual modes in the order of decreasing growth rate (“Lyapunov analysis”). Far away from the critical solution, no preferred gauge choice is known. We are then faced with the general question: Given initial data in general relativity, is there a prescription for the lapse and shift, such that, if these are in fact data for a self-similar solution, the resulting time evolution actively drives the metric to the special form (29) that explicitly displays the self-similarity? If such a prescription existed, one could try to find the non-linear critical solution itself as a fixed point of a renormalisation group transformation, as described by Bricmont and Kupiainen [15] for simple PDEs. Dolan’s [38] description of a RG flow as a Hamiltonian flow may be useful in making the identification. An incomplete answer to this question has been provided by Garfinkle [44]. His gauge conditions are ∂N 1 = N 2 K, ∂t 3

N i = 0,

(75)

where N is the lapse, N i the shift, and K the trace of the extrinsic curvature. Garfinkle now introduces the “scale-invariant” variables ˜ ik = N 2 hik , h

˜ ik = N −1 (Kik − Khik ), K

ωi = (ln N ),i .

(76)

N will turn out to absorb the overall spacetime scale. The introduction of ωi has the purpose of extracting the scale-invariant information from N . Garfinkle ˜ ik is gives an autonomous system of equations for these degrees of freedom (K traceless) alone. The trace K missing from this set is obtained by solving the Hamiltonian constraint for N K. The remaining degree of freedom, N , is evolved in time by Eqn. (75), but is not an active part of the autonomous system. If N K is periodic in t, we can write the spacetime metric in the form   ˜ ik dxi dxk , ˜ 2 dt2 + h (77) ds2 = e2Ct −N

R ˜ ∼ exp 1 (N K − hN Ki) dt, with where the constant C = 13 hN Ki, while N 3 hi the t-average. The constant C can be set equal to 1 by a rescaling of the coordinate t. In the notation of this review, t is then the scale coordinate τ . The Einstein equations have now been split into an autonomous scale-invariant ˜ ik , K ˜ ik ) and a scale variable (N ) which is driven passively by the scalepart (h invariant part. When the scale-invariant variables are periodic in t (independent of t), the spacetime is discretely (periodically) self-similar. It seems unlikely, however, that the converse holds: If we begin with data taken on an arbitrary hypersurface in a self-similar spacetime, Garfinkle’s evolution will not make his scale-invariant variables periodic in t. This should happen only if the initial 33

data have been collected on the “right” initial hypersurface. What is required is a gauge prescription that actively drives the time evolution towards a slicing that makes the scale-invariant variables periodic. Fixing the shift to be zero also poses a problem. In spherically symmetric homothetic spacetimes the homothetic vector becomes spacelike (inward pointing) at large radius on any spacelike slice. In Garfinkle’s ansatz the homothetic vector, if one exists, is identified with ∂/∂t, but with zero shift ∂/∂t is orthogonal to the hypersurfaces and therefore timelike, unlike the homothetic vector in the known examples of critical spacetimes. It seems necessary to allow for and prescribe a shift if Garfinkle’s scheme is to be consistent. In this context it may be relevant that in the split between the scale-invariant and scale part of the Einstein equations, the latter accounts only for one degree of freedom. This seems to indicate that any such split is tied to a slicing of spacetime. Garfinkle suggests that a dynamical explanation of critical phenomena would consist in finding a mechanism of “dissipation” that drives the scale-invariant system towards a fixed point or a limit cycle. One should keep in mind, however, that the limit cycle has at least one unstable mode, or else naked singularities would be generic.

5.2

Analytic approaches

A number of authors have attempted to explain critical collapse with the help of analytic solutions. The one-parameter family of exact self-similar real massless scalar field solutions first discovered by Roberts [86] has already been presented in section 4.6. It has been discussed in the context of critical collapse by Brady [13] and Oshiro et al. [80], and later Wang and Oliveira [94] and Burko [17]. The original, analytic, Roberts solution is cut and pasted to obtain a new solution which has a regular center r = 0 and which is asymptotically flat. Solutions from this family with p > 1 can be considered as black holes, and to leading order around the critical value p = 1, their mass is M ∼ (p− p∗ )1/2 . The pitfall in this approach is that only perturbations within the self-similar family are considered. But the p = 1 solution has many growing perturbations which are spherically symmetric (but not self-similar), and is therefore not a critical solution. This was already clear from the collapse simulations at the black hole threshold, but recently Frolov [43] has taken the trouble to calculate its perturbation spectrum explicitly (within spherical symmetry). The perturbations can be calculated analytically, and their spectrum is continuous, filling a sector of the complex plane, with Reλ ≤ 1. Soda and Hirata [88] generalize the Roberts solution to higher spacetime dimensions and calculate formal critical exponents. Oliveira and Cheb-Terrab [78] generalize the Roberts solution to the conformally coupled scalar field. Also in the context of critical phenomena in gravitational collapse, Koike and Mishima [72] consider a two-parameter family of solutions with a thin shell and an outgoing null fluid, and formally derive a “critical exponent”, but there 34

is no indication that these are critical solutions in the sense of having exactly one unstable mode. Husain [67] and Husain, Martinez and N´ un ˜ez [69] find perfect fluid “solutions” by giving a metric and reading off the resulting stressenergy tensor. The same authors [68] consider another scalar field exact solution (spherically symmetric with a conformal Killing vector, but not homothetic) and obtain a formal critical exponent of 1/2. Peleg and Steif [82] have analyzed the collapse of thin dust shells in 2+1 dimensional gravity with and without a cosmological constant. There is a critical value of the shell’s mass as a function of its radius and position. The black hole mass scales with a critical exponent of 1/2. By analogy with the Roberts solution, it is likely that in extending this mini-superspace model to a more general one, it would reveal itself not be an attractor of codimension one. Chiba and Soda [23] have noticed that a conformal transformation transforms Brans-Dicke gravity without matter into general relativity with a massless, minimally coupled scalar field. They transform the Choptuik solution from this so-called Einstein frame back to the physical, or Jordan, frame, and obtain a critical exponent for the formation of black holes in Brans-Dicke gravity that depends on the Brans-Dicke coupling parameter. The physical significance of this is doubtful in the absence of matter, as ω is defined in the Einstein frame only through the coupling of gravity to matter. [In the Einstein frame, all matter is minimally coupled to the Einstein metric times the conformal factor exp(ω + 32 )1/2 φ.] Oliveira [77] transforms not the Choptuik but the Roberts solution. Other authors have employed analytic approximations to the actual Choptuik solution. Pullin [84] has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin [83] have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a “transition edge” at constant selfsimilarity coordinate x. The nonlinearity of the gravitational field comes in through the matching procedure, and its details are claimed to provide an estimate of the echoing period ∆. While the insights of this paper are qualitative, some of its ideas reappear in the construction [50] of the Choptuik solution as a 1+1 dimensional boundary value problem. In summary, purely analytic approaches have remained surprisingly unsuccessful in dealing with critical collapse.

5.3

Astrophysical applications?

Any real world application of critical phenomena would require that critical phenomena are not an artifact of the simple matter models that have been studied so far, and that they are not an artifact of spherical symmetry. At present this seems a reasonable hypothesis. Critical collapse still requires a kind of fine-tuning of initial data that does not seem to arise naturally in the astrophysical world. Niemeyer and Jedamzik [76] have suggested a scenario 35

that gives rise to such fine-tuning. In the early universe, quantum fluctuations of the metric and matter can be important, for example providing the seeds of galaxy formation. If they are large enough, these fluctuations may even collapse immediately, giving rise to what is called primordial black holes. Large quantum fluctuations are exponentially more unlikely than small ones, P (δ) ∼ exp −δ 2 , where δ is the density contrast of the fluctuation. One would therefore expect the spectrum of primordial black holes to be sharply peaked at the minimal δ that leads to black hole formation. That is the required fine-tuning. In the presence of fine-tuning, the black hole mass is much smaller than the initial mass of the collapsing object, here the density fluctuation. In consequence, the peak of the primordial black hole spectrum might be expected to be at exponentially smaller values of the black hole mass than expected naively.

5.4

Critical collapse in semiclassical gravity

As we have seen in the last section, critical phenomena may provide a natural route from everyday scale down to much smaller scales, perhaps down to the Planck scale. Various authors have investigated the relationship of Choptuik’s critical phenomena to quantum black holes. It is widely believed that black holes should emit thermal quantum radiation, from considerations of quantum field theory on a fixed Schwarzschild background on the one hand, and from the purely classical three laws of black hole mechanics on the other (see [92] for a review). But there is no complete model of the back-reaction of the radiation on the black hole, which should be shrinking. In particular, it is unknown what happens at the endpoint of evaporation, when full quantum gravity should become important. It is debated in particular if the information that has fallen into the black hole is eventually recovered in the evaporation process or lost. To study these issues, various 2-dimensional toy models of gravity coupled to scalar field matter have been suggested which are more or less directly linked to a spherically symmetric 4-dimensional situation (see [46] for a review). In two space-time dimensions, the quantum expectation value of the matter stress tensor can be determined from the trace anomaly alone, together with the reasonable requirement that the quantum stress tensor is conserved. Furthermore, quantizing the matter scalar field(s) f but leaving the metric classical can be formally justified in the limit of many such matter fields. The two-dimensional gravity used is not the two-dimensional version of Einstein gravity but of a scalar-tensor theory of gravity. eφ , where φ is called the dilaton, in the 2dimensional toy model plays essentially the role of r in 4 spacetime dimensions. There seems to be no preferred 2-dimensional toy model, with arbitrariness both in the quantum stress tensor and in the choice of the classical part of the model. In order to obtain a resemblance of spherical symmetry, a reflecting boundary condition is imposed at a timelike curve in the 2-dimensional spacetime. This plays the role of the curve r = 0 in a 2-dimensional reduction of the spherically symmetric 4-dimensional theory. 36

How does one naively expect a model of semiclassical gravity to behave when the initial data are fine-tuned to the black hole threshold? First of all, until the fine-tuning is taken so far that curvatures on the Planck scale are reached during the time evolution, universality and scaling should persist, simply because the theory must approximate classical general relativity. Approaching the Planck scale from above, one would expect to be able to write down a critical solution that is the classical critical solution asymptotically at large scales, through an ansatz of the form ∞ X Z∗ (x, τ ) = enτ Zn (x), (78) n=0

where the scale L in τ = − ln(−t/L) is now the Planck length. This ansatz would recursively solve a semiclassical field equation, where powers of eτ (in coordinates x and τ ) signal the appearances of quantum terms. Note that this is exactly the ansatz (60), but with the opposite sign in the exponent, so that the higher order terms now become negligible as τ → −∞, that is away from the singularity on large scales. On the Planck scale itself, this ansatz would not converge, and self-similarity would break down. Addressing the question from the side of classical general relativity, Chiba and Siino [22] write down their own 2-dimensional toy model, and add a quantum stress tensor that is determined by the trace anomaly and stress-energy conservation. They note that the quantum stress tensor diverges at r = 0. This means that the additional quantum terms in the field equations carry powers not only of eτ , but instead of r−1 = x−1 eτ . Hence no self-similar ansatz can be regular at the center r = 0 (x = 0) even before the singularity appears at τ = ∞. They conclude that quantum gravity effects preclude critical phenomena on all scales, even far from the Planck scale. More plausibly, their result indicates that this 2-dimensional toy model does not capture essential physics. At this point, Ayal and Piran [4] make an ad-hoc modification to the semiclassical equations. They modify the quantum stress tensor by a function which interpolates between 1 at large r, and r2 /L2p at small r. The stress tensor would only be conserved if this function was a constant. The authors justify this modification by pointing out that violation of energy conservation takes place only at the Planck scale. It takes place, however, not only where the solution varies dynamically on the Planck scale, but at all times in a Planck-sized world tube around the center r = 0, even before the solution itself reaches the Planck scale dynamically. This introduces a non-geometric, background structure, effect at the world-line r = 0. With this modification, Ayal and Piran obtain results in agreement with our expectations set out above. For far supercritical initial data, black formation and subsequent evaporation are observed. With fine-tuning, as long as the solution stays away from the Planck scale, critical solution phenomena including the Choptuik universal solution and critical exponent are observed. (The exponent is measured as 0.409, indicating a limited accuracy of the numerical method.) In an intermediate regime, the quantum 37

effects increase the critical value of the parameters p. This is interpreted as the initial data partly evaporating while they are trying to form a black hole. Researchers coming from the quantum field theory side seem to favor a model (the RST model) in which ad hoc “counter terms” have been added to make it soluble. The matter is a conformally rather than minimally coupled scalar field. The field equations are trivial up to an ODE for a timelike curve on which reflecting boundary conditions are imposed. The world line of this “moving mirror” is not clearly related to r in a 4-dimensional spherically symmetric model, but seems to correspond to a finite r rather than r = 0. This may explain why the problem of a diverging quantum stress tensor is not encountered. Strominger and Thorlacius [89] find a critical exponent of 1/2, but their 2-dimensional situation differs from the 4-dimensional one in many aspects. Classically (without quantum terms) any ingoing matter pulse, however weak, forms a black hole. With the quantum terms, matter must be thrown in sufficiently rapidly to counteract evaporation in order to form a black hole. The initial data to be fine-tuned are replaced by the infalling energy flux. There is a threshold value of the energy flux for black hole formation, which is known in closed form. (Recall this is a soluble system.) The mass of the black hole is defined as the total energy it absorbs during its lifetime. This black hole mass is given by   12 δ (79) M≃ α where δ is the difference between the peak value of the flux and the threshold value, and α is the quadratic order coefficient in a Taylor expansion in advanced time of the flux around its peak. There is universality with respect to different shapes of the infalling flux in the sense that only the zeroth and second order Taylor coefficients matter. Peleg, Bose and Parker [81] study the so-called CGHS 2-dimensional model. This (non-soluble) model does allow for a study of critical phenomena with quantum effects turned off. Again, numerical work is limited to integrating an ODE for the mirror world line. Numerically, the authors find black hole mass scaling with a critical exponent of γ ≃ 0.53. They find the critical solution and the critical solution to be universal with respect to families of initial data. Turning on quantum effects, the scaling persists to a point, but the curve of ln M versus ln(p − p∗ ) then turns smoothly over to a horizontal line. Surprisingly, the value of the mass gap is not universal but depends on the family of initial data. While this is the most “satisfactory” result among those discussed here from the classical point of view, one should keep in mind that all these results are based on mere toy models of quantum gravity.

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6

Summary and conclusions

When one fine-tunes a one-parameter family of initial data to get close enough to the black hole threshold, the details of the initial data are completely forgotten in a spacetime region, and all near-critical time evolutions look the same there. The only information remembered from the initial data is how close one is to the threshold. Either there is a mass gap (type I behavior), or black hole formation starts at infinitesimal mass (type II behavior). In type I, the universal critical solution is time-independent, or periodic in time, and the better the fine-tuning, the longer it persists. In type II, the universal critical solution is scale-invariant or scale-periodic, and the better the fine-tuning, the smaller the black hole mass, according to the famous formula Eqn. (1). Both types of behavior arise because there is a solution which sits at the black hole-threshold, and which is an intermediate attractor. The basin of attraction is (at least locally) the black hole threshold itself, pictured as a hypersurface of codimension one that bisects phase space. Only the one perturbation mode pointing out of that surface is unstable. Depending on its sign, the solution tips over towards forming a black hole or towards dispersion. In the words of Eardley, all one-parameter families of data trying to cross the black hole threshold are funneled through a single time evolution. If the critical solution is timeindependent, its linear perturbations grow or decrease exponentially in time. If it is scale-invariant, they grow or decrease exponentially with the logarithm of scale. The power-law scaling of the black hole mass follows by a clever application of dimensional analysis. Although the mathematical foundations of this approach remain doubtful at present, it allows precise numerical calculation of the critical solution and the critical exponent, with good agreement with numerical “experiment”. The importance of type II behavior lies in providing a natural route from large to very small scales, with possible applications to astrophysics and quantum gravity. Natural here means that the phenomena persist for many simple matter models, without counterexample so far, and, apparently, even beyond spherical symmetry. As far as any generic parameter in the initial data provides some handle on the amplitude of the one unstable mode, fine-tuning any one generic parameter creates the phenomena. Moreover, scaling and echoing are seen already quite far from the threshold in practice. Clearly, more numerical work will be useful to further establish the generality of the mechanism, or to find a counter-example instead. In particular, future research should include highly non-spherical situations, initial data with angular momentum and electric charge, and matter models with a large number of internal degrees of freedom (for example, collisionless matter instead of a perfect fluid). Going beyond spherical symmetry, or including collisionless matter, will certainly pose formidable numerical challenges. The fundamental mathematical question in the field is why so many matter models (in fact, all models investigated to date) admit a critical solution, that 39

is, an attractor of codimension one at the black hole threshold. If the existence of a critical solution is really a generic feature, then there should be at least an intuitive argument, and perhaps a mathematical proof, for this important fact. While it is at present unclear if all matter models admit a self-similar solution with exactly one unstable mode(a type II critical solution), no reasonable matter model can admit a self-similar solution with no unstable modes, or naked singularities would be endemic in nature. Again one wonders if there is a general argument or proof for this fact. Progress in understanding general relativity as a dynamical system (in the presence of the slicing freedom) may be a crucial step on the way towards these proofs, and might also contribute to the study of singularities in general relativity. In the future, we can expect new phenomena based on continuous families of critical solutions. In numerical investigations we may also come across solutions on the black hole threshold with two or more unstable modes, although as a matter of terminology I would not call these “critical” solutions, because they would not “naturally” arise in collapse simulations. Critical solutions so far were scale-periodic or scale-invariant (type II), or static or periodic in time (type I). Are solutions conceivable which have neither of these symmetries, but which are still critical solutions in the essential sense of being attractors inside the black hole threshold? Numerical relativity has opened up a new research field in classical general relativity, critical phenomena in gravitational collapse. The interplay between numerical and analytic work in this new field is still continuing strongly. While its surprising features have captured the attention of many researchers in the GR community, it has also thrown some light on the outstanding problem of mathematical relativity, cosmic censorship. I would like to thank A. Abrahams, P. Bizo´ n, B. Carr, A. Coley, S. Jhingan and particularly A. Rendall for helpful comments on the draft paper.

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Figure 1: The phase space picture for discrete self-similarity. The plane represents the critical surface. (In reality this is a hypersurface of co-dimension one in an infinite-dimensional space.) The circle (fat unbroken line) is the limit cycle representing the critical solution. The thin unbroken curves are spacetimes attracted to it. The dashed curves are spacetimes repelled from it. There are two families of such curves, labeled by one periodic parameter, one forming a black hole, the other dispersing to infinity. Only one member of each family is shown.

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